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KTU, Dec 13, 2018@Kaiserslautern

Zhen-Sheng Yuan （中国科大 苑震生）

Atomic Spin Entanglement and Anyonic Statistics in Optical Lattices

USTC

University of Science and Technology of China

Z.-S. Yuan, KTU, DEC.13, 2018@Kaiserslautern

University of Science and Technology of China (USTC)

Frankfurt Beijing USTC, Hefei

Z.-S. Yuan, KTU, DEC.13, 2018@Kaiserslautern

University of Science and Technology of China (USTC)

USTC,Hefei Peking Shanghai

Z.-S. Yuan, KTU, DEC.13, 2018@Kaiserslautern

Z.-S. Yuan, KTU, DEC.13, 2018@Kaiserslautern

潘建伟 包小辉 陈凯 陈帅 陈宇翱 陆朝阳 徐飞虎 彭承志 苑震生 赵博 邓友金 张强 张军 刘乃乐 朱晓波 霍永恒 姚星灿 汪喜林 郁司夏 戴汉宁 陈腾云 江晓 印娟 任继刚 廖胜凯 李力

Jian-Wei Pan Shuai Chen Yu-Ao Chen Zhen-Sheng Yuan Bo Zhao You-Jin Deng Xing-Can Yao Hanning Dai

Z.-S. Yuan, KTU, DEC.13, 2018@Kaiserslautern

Introduction to our team

Research field: quantum information processing with photons and atoms  Quantum communication

Metropolitan fiber quantum communication networks Quantum memory and quantum repeater Free space quantum communication

Z.-S. Yuan, KTU, DEC.13, 2018@Kaiserslautern

Introduction to our team

Research field: quantum information processing with photons and atoms  Quantum computation and simulation with

Multi-photon entanglement Superconducting qubit Atom-atom entanglement Ultracold Bose gases (SOC) Ultracold Fermion mixture Ultracold molecule

Z.-S. Yuan, KTU, DEC.13, 2018@Kaiserslautern

Motivation: Quantum Information Processing

Resource for QIP, Entangled states

Ions: Monz et al, PRL 106, 130506 (2011); N. Friis et al, PRX 8, 021012 (2018); J Zhang et al, Nature 551, 601 (2017) Photons: X-L Wang et al, PRL 117, 210502 (2016); arXiv:1801.04043 Superconducting qubits: P. Roushan et al, Science 358, 1175 (2017) Google;

• N. Kalb et al, Science 356, 928 (2017), intel Qutech; IBM 49 qubits; Yale;

Ions: R. Blatt, C. Monroe Photons: Jian-Wei Pan Superconducters: Google, IBM, Intel

Z.-S. Yuan, KTU, DEC.13, 2018@Kaiserslautern

Scalability: atoms in optical lattice

Optical lattice: an array of well coherently controlled cold atoms in-situ imaging: only one atom trapped in a lattice Spin exchange interaction: generate spin-spin entanglement

Multi-atom entanglement!

Vaucher et al, NJP (2008)

Z.-S. Yuan, KTU, DEC.13, 2018@Kaiserslautern

Scalability: fault tolerable qubits

To overcome qubit errors in quantum computing

 Error-correcting code

• Shor, PRA 52, R2493 (1995) 9qubits
• Steane, PRL 77, 793 (1996) 7qubits
• Laflamme et al., PRL 77, 198 (1996) 5qubits

 Traditional concatenated codes require error rate < 210-5 !  Protect quantum bits/gates at the physical level -- topological quantum computing

• Kitaev, Ann. Phys. 303, 2 (2003); Ann. Phys. 321, 2 (2006)
• Raussendorf et al., Ann. Phys. 321, 2242 (2003)
• Nayak et al., RMP 80 (3): 1083 (2008)

 Relax the error threshold rate from 10-5 to 10-2

Z.-S. Yuan, KTU, DEC.13, 2018@Kaiserslautern

Scalability: fault tolerable qubits

Topological Quantum Computation

Quantum gates--Braiding Anyons Protect qubits with energy gap

Anthony James Leggett: …no naturally occurring system is

likely to have a Hamiltonian (for topological computing); Purpose- engineered systems of optical lattices or Josephson junction arrays (are promising candidates)

Ground states Excited states ∆𝐹 Energy Gap ۧ |𝜔

Z.-S. Yuan, KTU, DEC.13, 2018@Kaiserslautern

Kitaev Model: Toric code

Kitaev, Annals of Physics 303, 2 (2003)

• Four-body interaction
• Abelian Anyons: e, m excitaions

Protecting qubits with energy gap Hamiltonian:

Z.-S. Yuan, KTU, DEC.13, 2018@Kaiserslautern

Toric code -- Braiding 𝜏

𝑘 𝑨

e e

𝜏

𝑘 𝑌

m m

𝜏

𝑘 𝑌

m m

Z.-S. Yuan, KTU, DEC.13, 2018@Kaiserslautern

e e

𝜏

𝑘 𝑌

m m Topological phase 𝑓𝑗𝜚, 𝜚 = 𝜌 No e-excitation, 𝜚 = 0 1 2 3 4

𝜏1

𝑌

m m

𝜏4

𝑌

m m

𝜏3

𝑌

m m

𝜏2

𝑌

m m ൿ ห𝜒′ = 𝑓𝑗𝜚| ۧ 𝑓, 𝑓, 𝑛, 𝑛 ۧ |𝜒 = | ۧ 𝑓, 𝑓, 𝑛, 𝑛

Toric code -- Braiding

Z.-S. Yuan, KTU, DEC.13, 2018@Kaiserslautern

Spin entanglement and anyonic statistics in OL

Our experiment:

• Manipulating superexchange in optical lattice
• Creating entangled atom pairs
• Manipulating four-body interaction, four-atom entanglement
• Demonstrating anyonic statistics with plaquette units

Entangled atom pairs Ring exchange and Toric code

Z.-S. Yuan, KTU, DEC.13, 2018@Kaiserslautern

Atoms in optical lattices

Bose-Hubbard model (BHM) J: nearest-neighbor tunneling 𝑉: onsite interactions J U Standing wave of light 3D optical lattice

Z.-S. Yuan, KTU, DEC.13, 2018@Kaiserslautern

Experimental setup

Vacuum chamber MOT BEC Magnetic Transfer BEC

Z.-S. Yuan, KTU, DEC.13, 2018@Kaiserslautern

11 degree

Lat-Y

Pancake-2

• Objective: NA=0.48, resolution 2 μm

Prepare a 2D quantum gas with in-situ imaging

87Rb:

ۧ |𝐺 = 1, 𝑛𝐺 = −1 BEC 2 × 105 atoms Load into a pancake trap 𝑂2D~15000, 𝑈2D=23(3) nK SF to MI transition by ramping up lattice depth

Z.-S. Yuan, KTU, DEC.13, 2018@Kaiserslautern

Optical super-lattice

Theory: Duan et al., PRL 91, 090402 (2003) Experiment: Trotzky et al., Science 319, 295 (2008)

Isolated double wells: 𝑊 𝑦 = 𝑊

𝑡 cos2 2𝑙𝑦 + 𝜚𝑦 + 𝑊 𝑚 cos2 𝑙𝑦

| ۧ ↑ = 5𝑇 Τ

1 2|

ۧ 𝐺 = 2, 𝑛𝐺 = −2 | ۧ ↓ = 5𝑇 Τ

1 2|

ۧ 𝐺 = 1, 𝑛𝐺 = −1

Z.-S. Yuan, KTU, DEC.13, 2018@Kaiserslautern

Spin super-exchange: generating spin entanglement 𝐾𝑓𝑦~ 4𝐾2/𝑉

Interaction dominated 𝑉 ≫ 𝐾 , with pseudo spins: | ۧ ↓↑ Initial state:| ۧ ↑↓ is degenerate with The spins will oscillate between the two configurations with a period of 1/𝐾𝑓𝑦 Stop the oscillation by increasing the barrier to create spin entanglement

1 2 |

ۧ ↑↓ + | ۧ ↓↑

Z.-S. Yuan, KTU, DEC.13, 2018@Kaiserslautern

Spin-dependent superlattices

Normal super-lattice

VS Vl

+

Spin-dependent superlattice

angle between two polarization planes of laser S

| ۧ ↑ = 5𝑇 Τ

1 2|

ۧ 𝐺 = 2, 𝑛𝐺 = −2 , 𝑕𝐺 = Τ 1 2 | ۧ ↓ = 5𝑇 Τ

1 2|

ۧ 𝐺 = 1, 𝑛𝐺 = −1 , 𝑕𝐺 = Τ −1 2

Left well is higher Right well is higher

𝛼𝐶

Z.-S. Yuan, KTU, DEC.13, 2018@Kaiserslautern

Spin-dependent superlattices

Z.-S. Yuan, KTU, DEC.13, 2018@Kaiserslautern

π pulse, ωL

ωL ωR effective magnetic gradient caused by spin-dependent superlattice B1 B2 B1 B2

Spin-dependent superlattices

Z.-S. Yuan, KTU, DEC.13, 2018@Kaiserslautern

• Increase 𝑊

𝑗  Freeze entangled state

𝑊

𝑗

How to detect entanglement?

• Switch off effective magnetic gradient, |↑↓ and |↓↑ degenerate
• Decrease 𝑊

𝑗  spin oscillation

J/U=0.11, decay 120ms Vs=16Ers, Vl=40Erl | = 4𝐾2 𝑉

Spin super-exchange: generating spin entanglement

Z.-S. Yuan, KTU, DEC.13, 2018@Kaiserslautern

Spin-dependent collisional loss: identify |↓↓ from 4 spin basis

Entanglement detection

 Imaging spin-up atoms  Count N1

 π pulse

 Merging and killing  Count N2

Identify | ↑↓ , | ↓↑ , |↑↑: transfer to |↓↓ by left/right π pulse

𝑂↓↓ = 𝑂𝑈𝑝𝑢𝑏𝑚 − 𝑂

1 − 𝑂2

Entangled state: 𝜔 =

1 2 ( ↑↓ + ↑↓ )

Z.-S. Yuan, KTU, DEC.13, 2018@Kaiserslautern

Detection of entanglement Spin correlation curve

Violation of CHSH type Bell’s inequality S = 2.21± 0.08 Dai et al., Nature Physics 12, 783 (2016)

Z.-S. Yuan, KTU, DEC.13, 2018@Kaiserslautern

isolated plaquettes

2D-optical superlattice

B.Paredes & I.Bloch, PRA77,23603 (2008).

𝐈=𝑲฀𝑻𝟐𝑻𝟑𝑻𝟒𝑻4

BHM： Super-exchange: Ring-exchange:

J<<U

Z.-S. Yuan, KTU, DEC.13, 2018@Kaiserslautern

Ring-exchange interaction 𝐵𝑡 = −𝜏1

𝑦𝜏2 𝑦𝜏3 𝑦𝜏4 𝑦

4th order perturbation to the BHM

෡ 𝐼(4) = 40 ൗ 𝐾4 𝑉3

~Hz

Z.-S. Yuan, KTU, DEC.13, 2018@Kaiserslautern

2D-optical superlattice

𝐈=𝑲฀𝑻𝟐𝑻𝟑𝑻𝟒𝑻4

BHM ： Super-exchange: Ring-exchange:

J=200 Hz, U=2 kHz 𝐾𝑓𝑦 ∼

𝐾2 𝑉 = 20 Hz ~ 1 nK

𝐾฀ ∼

𝐾4 𝑉3 = 0.2 Hz ~ 0.01 nK

Z.-S. Yuan, KTU, DEC.13, 2018@Kaiserslautern

Minimum toric code Hamiltonian

Toric code model in subspace ring exchange

Degenerate

Z.-S. Yuan, KTU, DEC.13, 2018@Kaiserslautern

Spectrum of the plaquette model

Z.-S. Yuan, KTU, DEC.13, 2018@Kaiserslautern

Site-resolved addressing: state initialization

Effective magnetic gradient created by the spin-dependent superlattices Sawtooth-like, period of OL

𝐶3 > 𝐶4 = 𝐶2 > 𝐶1 𝐶

3 1 2 4 1 2 3 4

Z.-S. Yuan, KTU, DEC.13, 2018@Kaiserslautern

Ring Exchange Driven Oscillation

= 1 2 ۧ |𝐵− + ۧ |𝐵+ Initial state

Z.-S. Yuan, KTU, DEC.13, 2018@Kaiserslautern

Observation of ring exchange driven oscillation

Count the populations of different states

𝜌 pulse 𝜌 pulse Imaging, Dark Imaging, Bright

Z.-S. Yuan, KTU, DEC.13, 2018@Kaiserslautern

Topological phase of Abelian anyons,  = /2

e m m m m m m m m e m m m m e m m m e m m m e m m m m m m m m m m m m m m m m m m

ۧ |𝐵−

𝜏4

𝑦

𝜏4

𝑦

ۧ −|𝐵− ۧ +|𝐵+

𝜏3

𝑦

𝜏3

𝑦

𝜏2

𝑦

𝜏2

𝑦

𝜏1

𝑦

𝜏1

𝑦

𝜔𝑔 = 1 2 ۧ − 𝑗|𝐵− + ۧ |𝐵+ 𝜔𝑗 = 1 2 ۧ 𝑗|𝐵− + ۧ |𝐵+

ۧ |𝐵+

Z.-S. Yuan, KTU, DEC.13, 2018@Kaiserslautern

Observation of Anyonic Fractional Statistics

Dai et al, Nature Physics 13, 1195 (2017)

𝟐 𝟑 ۧ − 𝒋|𝑩− + ۧ |𝑩+ 𝟐 𝟑 ۧ 𝒋|𝑩− + ۧ |𝑩+

| ۧ ↓↑↓↑ | ۧ ↑↓↑↓

Z.-S. Yuan, KTU, DEC.13, 2018@Kaiserslautern

Outlook- towards large entangled state

 For a large entangled state: remove defects, connect the atom pairs Challenge: cool the atoms in lattices?

Cooling in Lattices ??

Z.-S. Yuan, KTU, DEC.13, 2018@Kaiserslautern

Outlook- towards large entangled state

 Challenge theoreticians at an unprecedented level  High-resolution imaging system Numerical aperture: NA=0.8; Resolution: 690 nm

Z.-S. Yuan, KTU, DEC.13, 2018@Kaiserslautern

Outlook- simulating topological materials

 Couple the plaquettes; strongly correlated topological system  Extend to fermionic systems; non-Abelian ...

Theo: CW Zhang et al, PNAS (2007) Reviews on topological matters with ultracold atoms: Goldman, Budish&Zoller, Nat. Phys. (2016) Zohar et al. Rep. Prog. Phys. (2016)

Z.-S. Yuan, KTU, DEC.13, 2018@Kaiserslautern

Shaking, spin dependent shaking

X direction Y direction

Z.-S. Yuan, KTU, DEC.13, 2018@Kaiserslautern

The team members

Jian-Wei Pan Han-Ning Dai Bing Yang Andreas Reigruber Xiaofan Xu Yu-Ao Chen Hui Sun USTC

• Univ. Heidelberg